Change of Base Formula for Logarithms
Change of Base Formula converts a logarithm from one base to another base. The formula expresses logᵦ x as a quotient of logarithms with any valid common base. This page explains the formula, base restrictions, reciprocal property, calculation method and numerical examples used in simplification questions.
What Is the Change of Base Formula?
Here, x > 0, b > 0, b ≠ 1, a > 0 and a ≠ 1. The new base a can be any valid base, commonly 10 or e. For example, log₂ 8 = log₁₀ 8 ÷ log₁₀ 2 = 3 because 2³ = 8.
Change of Base Formula Formula & Tricks
Important Formulas
Use any base a that is positive and not equal to 1. Both logarithms on the right must have the same base.
This form uses base 10 logarithms, which are called common logarithms.
This form uses natural logarithms with base e.
Interchanging the number and the base gives reciprocal logarithms, provided both logarithms are defined.
Quick Tricks
Select base 10 or base e for direct calculator evaluation. If the numbers are powers of a common number, choose that number to simplify mentally.
If a required logarithm has its base and argument reversed, take the reciprocal instead of recalculating both logarithms.
Change of Base Formula Concepts
Conditions for Valid Base Conversion
In the change of base formula, the new base a must also satisfy a > 0 and a ≠ 1. The denominator logₐ b cannot be zero because logₐ 1 = 0, so b must not be 1.
Conversion Using Base 10 or Base e
The same type of logarithm must be used in the numerator and denominator. Thus, log₅ 25 = log 25/log 5 = 2, and the natural-log form gives the same result: ln 25/ln 5 = 2.
Changing Between Any Two Bases
The selected base does not change the value of the logarithm; it only changes the form used for calculation. This is useful when the original base is not directly available for evaluation.
Reciprocal and Product Properties
A related product property is log_a b × log_b c = log_a c. Both results follow directly from the change of base formula and require valid logarithms.
Change of Base Formula Video Lessons
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Logarithms: Change of Base and Expressions
Understand the change-of-base formula for logarithms and learn how to simplify and evaluate logarithmic expressions commonly used in quantitative aptitude problems.
Practice Change of Base Formula Questions
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Change of Base Formula: Quick Revision
Use these formulas and restrictions while solving logarithm conversion questions.
- log_b x = logₐ x/logₐ b.
- The common base a must be positive and not equal to 1.
- For log_b x, x must be positive, b must be positive and b ≠ 1.
- log_b x = log₁₀ x/log₁₀ b = ln x/ln b.
- log_b a = 1/log_a b.
- log_a b × log_b c = log_a c.
- Use the same logarithm base in the numerator and denominator.
Change of Base Formula FAQs
What is the change of base formula for logarithms?
The formula is log_b x = logₐ x/logₐ b, where a is any positive base other than 1.
How do you calculate log₂ 8 using the change of base formula?
log₂ 8 = log₁₀ 8/log₁₀ 2 = 3, because 2³ = 8.
Can the new base in the formula be any number?
It can be any positive number other than 1. For example, base 10 and base e are commonly used.
How is log₄ 8 evaluated by changing the base to 2?
log₄ 8 = log₂ 8/log₂ 4 = 3/2.
What is the reciprocal of log₂ 7?
The reciprocal is log₇ 2, because log₂ 7 × log₇ 2 = 1.
Why must the numerator and denominator use the same base?
The change of base identity is derived by expressing both logarithms relative to one common base. Therefore, logₐ x and logₐ b must have the same base a.